Preheat an electric aebleskiver pan according to the manufacturer's directions.

Beat together the eggs, buttermilk, and butter in a mixing bowl until smooth. Gradually whisk the flour, salt, sugar, baking soda, and cardamom into the egg mixture until evenly blended and smooth.

Spoon 1 tablespoon of shortening into each cup of the preheated aebleskiver pan, and allow the oil to heat. Pour batter into each cup until 3/4 full. Allow the batter to cook in the bottom of the cup 3 to 4 minutes until golden brown, then use a long skewer or knitting needle to lift one edge and give it a 1/4 turn. The uncooked batter will spill over into the cup, and cook. Continue lifting and turning the edges until the batter is fully cooked into a golden brown ball. Drain on paper towels.